Heard D.E. (editor) Analytical Techniques for Atmospheric Measurement
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192 Analytical Techniques for Atmospheric Measurement
or approximating e
−x
≈ 1 −x,
R = Ecl
T Pd (4.3)
where E is the excitation power in photons/s and the term inside the brackets of
Equation (4.2) is the fraction of photons absorbed by the analyte of number density c
over the path length l. The integral represents the overlap between the spectral profile
of the excitation source and the absorption cross section of the molecule ,
where is frequency. This equation assumes the fraction of molecules in the excited
state to be small. Use of the full Einstein equations for stimulated absorption, stimulated
emission, and spontaneous emission is required if populations approach 50%. Under such
conditions the transition is said to be saturated and the fluorescence is independent of
input power. Since most fluorescence instruments are limited by noise, which is linearly
proportional to the power of the light source, saturation is not advantageous.
The fluorescence quantum yield, Q, is the fraction of excited molecules that fluoresce,
that is (photons emitted)/(photons absorbed). For molecules that can be treated as two-
level systems in which the only means of relaxation from the excited state to the ground
state are radiative or collisionally induced, the fluorescence quantum yield is given by
Q =
k
rad
k
rad
+
i
k
qi
M
i
(4.4)
where k
rad
is the radiative rate constant of the excited species 1/
R
and k
qi
are the
fluorescence quenching rate constants due to collisional deactivation by bath molecules
M
i
, mainly nitrogen, oxygen, water, argon and carbon dioxide. Table 4.1 summarizes
these relevant parameters for compounds commonly measured by fluorescence.
Often the quenching rate k
qi
M
i
is much greater than k
rad
. Under these circumstances
the 1/[M] term in the fluorescence quantum yield combines with the c term in the
excitation rate to make the overall signal rate proportional to c/M, which is the mixing
ratio. This allows the detection cell to be held at lower than ambient pressure without a
penalty in the fluorescence signal, as the negative effect of the decreased number density
is nearly equally offset by an increase in the fluorescence quantum yield due to decreased
quenching. For example, the (0–0) transition of OH at 308 nm has a radiative lifetime
of 676 ns (Bailey et al., 1997), and k
qi
M
i
is ten times this radiative rate at 7 torr. For
NO
2
and NO
3
, which have excited state lifetimes of ∼100s, the quenching rate is faster
than the rate of radiative decay at pressures well below 1 torr.
The collection efficiency, C, is given by
C =
Td
t
2
t
1
exp
−t
dt (4.5)
where represents the fraction of the solid angle of the fluorescence intercepted by the
collection lens and focused onto the photocathode; the first integral represents the trans-
mission through the collection optics T, the quantum efficiency of the PMT and
the emission spectrum of the molecule ; and the second integral is the fraction of
the fluorescence which occurs in the detection time gate of width t
2
−t
1
for an excited
state lifetime .
Fluorescence Methods 193
Table 4.1 Relevant spectroscopic parameters of a few compounds measured by fluorescence
Species Absorption cross section
(base e) (cm
2
)
Excited state
lifetime
0
Quenching rate
constants k
q
cm
3
molecule
−1
s
−1
Additional
compounds
measurable
Cl 39 ×10
−13
2.6 ns ClO, ClONO
2
,
(118.0 nm) (1) ClOOCl
OH 14 ×10
−15
(282 nm) 676 ns (3)
˜
A
2
+
= 0:HO
2
,H
2
O
71 ×10
−15
(308 nm) (2) 33×10
−11
N
2
14 ×10
−10
O
2
(3)
NO 19 ×10
−17
cm
2
(4) 205 ns (5)
˜
A
2
= 0:NO
2
,NO
y
(single-photon) 37 ×10
−11
N
2
15 ×10
−10
O
2
(6)
NO
2
59 ×10
−19
408 nm 65–130 s (8) 6 ×10
−11
N
2
Peroxy nitrates,
14 ×10
−20
640 nm (7) 54 ×10
−11
O
2
excited
= 532 nm
alkyl and
hydroxy-alkyl
nitrates, HNO
3
NO
3
22 ×10
−17
662 nm biexponential, B
2
E
= 0000:N
2
O
5
15 ×10
−17
623 nm (9) 3–30 s fast, 17 ×10
−11
N
2
340 s slow (10, 11) 21×10
−11
O
2
(11)
HCHO 16×10
−20
(354 nm) (12) 50 ns–4 s (13, 14) 52×10
−13
N
2
27 ×10
−12
O
2
excite
=3532 nm (15)
SO
2
158 ×10
−17
13–600 s (17, 18) 145 ×10
−11
(He)
excite
= 20356 nm (16)
excite
= 266 nm (19)
Sources: (1) Anderson et al., 1980; (2) McGee & McIlrath, 1984; (3) Bailey et al., 1997; (4) Bradshaw et al., 1982;
(5) Luque & Crosley, 2000; (6) Nee et al., 2004; (7) Voigt et al., 2002; (8) Patten et al., 1990; (9) Yokelson et al.,
1994; (10) Ishiwata et al., 1983; (11) Nelson et al., 1983; (12) Cantrell et al., 1990; (13) Moore & Weisshaar, 1983;
(14) Miller & Lee, 1978; (15) Shibuya & Lee, 1978; (16) Martinez & Joens, 1992; (17) Holtermann et al., 1980; (18) Brus
& McDonald, 1974; (19) Zhang et al., 1998.
Although detailed knowledge of each of these components of the signal is essential to
predict the performance of a fluorescence measurement (Bloss et al., 2003, Holland et al.,
1995, Stevens et al., 1994, Wennberg et al., 1994a, Thornton et al., 2000), in practice
most of these quantities do not need to be explicitly evaluated. Rather, laser-induced
fluorescence (LIF) instruments are calibrated using a known concentration of the sampled
compound. Calibration is also essential because of potential sampling losses.
4.2.3 The background
The background (or zero) in a fluorescence instrument typically consists of
B =B
chamber
+B
P-
scatter
+B
solar
+B
PMT
+B
interference
(4.6)
where B
chamber
is optical scatter from surfaces within the optical cell; B
P-scatter
is the sum of
pressure-dependent Raman, Rayleigh and Mie scattering; B
solar
is solar-induced; B
PMT
is
194 Analytical Techniques for Atmospheric Measurement
from the dark current of the PMT; and B
interference
represents any potentially time-varying
signal, for example fluorescence or scattering by other atmospheric gases or aerosol.
General strategies for reducing the background of a fluorescence measurement include
use of geometric baffles to block scattered laser light, operation at low pressure to reduce
B
P-scatter
and to enable time-gated detection, and use of optical filters.
The background in fluorescence instruments is commonly measured by one or more of
the following methods: introducing zero air to the instrument, selectively scrubbing the
sampled air of the targeted compound, or tuning the excitation frequency off-resonance
with the chosen absorption. As B inevitably varies for reasons that are inscrutable, it is
advantageous to make S B. Indeed the success of many fluorescence instruments stems
from the near-zero backgrounds achieved.
4.2.4 The signal-to-noise ratio and detection limit
The mixing ratio is related to the fluorescence signal S by
=
S
(4.7)
where is the instrument’s calibration constant in counts/s/ppb, S is experimentally
determined by subtracting the background count rate B from the observed gross signal
S +B. The associated uncertainty,
S
, with this subtraction is given by:
S
=
2
S+B
+
2
B
(4.8)
where
S+B
and
B
are the uncertainties of the total count rate and background count
rates, respectively. The noise
X
associated with a photon count rate X over an interval
of time t is equal to Xt
1/2
. Thus if equal amounts of time are spent measuring B and
S,
S
is equal to St +2Bt
1/2
and the signal-to-noise ratio, SNR, can be expressed as
SNR =
St
S
=
St
√
St +2Bt
=
S
√
t
√
S +2B
(4.9)
Note that if the background and signal are proportional to excitation power, then the
SNR will be proportional to the square root of the excitation power and to the square
root of the averaging time.
At the detection limit, S 2B. Combining Equations 4.7 and 4.9 gives the minimum
detectable mixing ratio,
min
:
min
=
SNR
2B
t
(4.10)
If the uncertainty in B is negligible because it is not varying or only varying slowly with
time, then the time spent measuring B can be reduced to a negligible fraction of the total
measurement time for an additional factor of
√
2 in the SNR. This is often the case for
instruments where B is proportional to laser power or determined by thermally induced
dark counts. Although longer averaging times can eliminate shot noise, pure statistical
Fluorescence Methods 195
noise rarely sets the ultimate detection limit on instrument performance. Factors such
as Mie scattering from a time-varying atmospheric aerosol concentration, fluorescence
of other gases, instability of the laser power, laser wavelength or laser alignment often
determine the fundamental limit that cannot be improved upon by signal averaging.
4.2.5 General design of fluorescence instruments
At the heart of a fluorescence instrument is the optical cell, in which three elements
intersect: gas flow, excitation and detection. Typical designs orient these three quantities
along orthogonal axes so that the fluorescence is observed against a dark background.
Figure 4.2 shows a typical detection cell. Laser light enters and exits a cell through
Brewster-angle windows and passes through a series of narrow aperture baffles to reduce
scattered laser light. Fluorescence emanating from the center of the cell is collected
and collimated by a lens, passes through an optical filter and is then focused onto the
photocathode of a PMT by a second lens.
Lamps have been successfully used for numerous fluorescence measurements, including
O
3
P (Anderson, 1975; Kita et al., 1996), halogens (Anderson et al., 1980), sulphur
dioxide (Luke, 1997), carbon monoxide (Holloway et al., 2000), and OH measurements
(Anderson, 1976). However, the use of lasers offers many advantages:
1) Lasers generally offer more intense beams of light.
2) Pulsed lasers allow for time-gated detection methods which greatly reduce background
noise.
3) Scattered light can be minimized by aligning the electric field vector of the laser light
parallel to the detection axis, as all types of scattering (Raman, Rayleigh, etc.) are
primarily emitted in the plane perpendicular to the electric field vector.
4) Laser light can be relatively-easily multi-passed through an optical cell.
5) The tunability and narrow linewidths of some lasers allow for spectral modulation as
a test of interferences or for measurement of backgrounds.
Time gating is most effective when the detection cell is held at a pressure low enough
that the excited state lifetime is much longer than the duration of the laser pulse. Delayed
detection of fluorescence is then nearly free of laser-induced background. The application
of low-pressure time-gated fluorescence, techniques which had been frequently used in
the laboratory, for atmospheric measurements was pioneered by Thomas Hard, Robert
O’Brien and co-workers at Portland State University and they named it FAGE: fluores-
cence assay by gas expansion (Hard et al., 1984). In FAGE, sampled air is expanded
through a critical orifice into a low-pressure detection cell. In addition to the reduction
in background attained by temporal rejection of the laser-induced background, FAGE
offers the following advantages:
1) Pressure-dependent sources of background noise (Rayleigh, Raman and Mie scattering)
are reduced.
2) The decrease in pressure broadening allows for spectral modulation of the fluorescence
signal using the rotational structure of the analyte’s absorption spectrum.
3) The reduced gas number density minimizes chemical and photolytic interferences.
196 Analytical Techniques for Atmospheric Measurement
Laser in
out
Fluorescence
detection
region
Figure 4.2 A single-pass fluorescence cell. Laser light enters and exits the cell through Brewster-angle
windows and passes through a series of baffles. A fraction of the isotropically emitted fluorescence from
the detection region is collected and collimated by a set of lenses, passed through a spectral filter, and
is then focused onto the photocathode of a PMT by a second set of lenses. Gas flow is perpendicular to
the page. (From Kanaya et al., 2001, with kind permission of Springer Science and Business Media.)
Fluorescence Methods 197
The disadvantages of the low-pressure technique are the need for vacuum pumps and
potential surface losses on the pressure-reducing critical orifice of the inlet. There is
also a fall-off in the calibration constant at pressures low enough that the reduction in
quenching rate no longer offsets the reduced number density.
Figure 4.3 illustrates the time gates used for detection of NO
2
at a cell pressure of
2 torr. The fluorescence lifetime of NO
2
at this pressure is approximately 270 ns. The
laser-induced noise is almost entirely coincident with the 40 ns transit time of the laser
pulse in the detection cell. Thus an 800 ns wide detection gate, delayed by 55 ns, reduces
the background by 95% while collecting ∼75% of the fluorescence. With a repetition
rate of 8 kHz, this gives an experimental duty cycle of 0.64%, reducing any continuous
noise sources (e.g. B
solar
B
PMT
) by a factor of 156. To synchronize the excitation pulse
with the detection electronics, a beamsplitter diverts a small fraction of the laser beam
and directs it to a fast photodiode. The signal from this photodiode serves as the trigger
for the photon counting electronics.
A single-pass design was used for the instrument shown in Figure 4.2; however, many
fluorescence instruments use a multi-pass design utilizing highly reflective >999%
mirrors. The obvious advantage of multi-passing is the increased excitation rate produced
by the elevated photon flux. The price of this enhancement is usually an increased
background resulting from light scattered by the mirrors themselves. This scattered
light is often red-shifted, diffuse and emanates from a large area, making it difficult to
completely eliminate using geometric baffles. The most common multi-pass design is a
White cell. Herriott cells (Herriott et al., 1964) are also used (Cleary et al., 2002; Wood
et al., 2003), though these are less common due to the increased likelihood of saturation
250
225
200
175
150
125
100
75
50
25
0
0 255075100
Gate delay (ns)
Counts /second
Delayed channel gate
Prompt channel gate
125 200 400 600 800
Figure 4.3 Reduction of the background by time gated detection. Total signal for 50 ppb NO
2
(squares)
and 0 ppb NO
2
(circles), i.e., background signal, at 2 torr are plotted versus delay from when the laser
pulse enters the detection cell. One channel of the counter catches all signal prompt with the laser pulse
and has an 800 ns wide gate. The other channel has a 55 ns delayed, 800 ns wide gate to eliminate scatter
prompt with the laser pulse. The area under the fluorescence curve remains large ∼75% in the delayed
gate, while the background noise has essentially decayed to zero. (From Thornton et al., 2000, reprinted
with permission from American Chemical Society.)
198 Analytical Techniques for Atmospheric Measurement
of the electronic transition as each pass of the laser light is focused into the centre of
the cell. A spherical mirror, placed at a distance of its radius of curvature below the
centre of the optical cell, is often used to double the solid angle collected. As illustrated
in Figure 4.2, baffles are used to shield the detector optics from scattered laser light.
Additional baffles are often placed on the outer edge of the collection lens and retro-
reflector. These baffles and the interior of detection cells are often painted with a low
fluorescence optical black paint.
Optical filters are used to transmit the desired fluorescence in a certain spectral region
while rejecting all other light. Fluorescence can be resonant with the excitation wavelength,
as for OH detection at 308 nm; red-shifted from the excitation wavelength, as for NO
2
detection; or blue-shifted, as for two-photon LIF detection of NO. Bandpass filters, long-
pass filters, short-pass filters or some combination of these three are used. The photon
flux of 100 mW of 532 nm light is 27 ×10
17
photons/s. With this light source, the
Rayleigh-scattered photon flux from nitrogen and oxygen in a typical multi-pass optical
cell at 2 torr is about 10
10
photons/s, and the magnitude of chamber scatter is usually
comparable. Typically about 1% of the scatter is imaged onto a photocathode with a
quantum efficiency of 10%. Thus attenuation of order 10
−7
is desirable to completely
reject this scattering.
The main types of optical filters used are coloured glass filters, liquid chemical solution
filters and interference filters. The first two are based on absorption, while interference
filters have dielectric coatings which reject light by reflection. Coloured glass filters
have sharp cut-on wavelengths and high transmissions in their spectral pass. Increased
attenuation can be attained by use of multiple absorbing filters. The major drawback of
absorbing filters is that the filters themselves fluoresce (Fong & Brune, 1997; Pfeiffer &
Porter, 1964). Liquid filters (Anderson, 1976; Bradshaw et al., 1985; Fong & Brune,
1997) contain a concentrated chemical solution with a distinct absorption spectrum
contained between two solid windows. Interference filters exhibit high attenuation and
high transmission and when coated with high grade quartz usually do not fluoresce;
however, use of multiple filters does not lead to increased attenuation. Additionally, the
cut-on wavelength of interference filters depends on the angle of incidence, requiring
geometric baffling for rejection of scattered light that has sources in a different physical
location than the desired molecular or atomic fluorescence. Detection of resonance
fluorescence precludes such filtering (besides the use of bandpass filters) and requires
efficient temporal gating as previously described, in order to reject laser scatter and
Rayleigh scattering.
The overview above provides an introduction to the design criteria for measurement
of atmospheric trace gases by fluorescence. In the following sections we discuss specific
examples. Issues related to sampling which were given limited attention above but which
are usually as challenging as detection will be discussed in more detail.
4.3 The hydroxyl radical
The hydroxyl radical, OH, regulates the concentration of the majority of trace gases,
including ozone (Wennberg et al., 1994b), and defines the oxidative capacity of the
Fluorescence Methods 199
atmosphere (Levy, 1972; Logan et al., 1981). Measurement of OH is difficult: peak concen-
trations in the troposphere and stratosphere are in the order of 10
6
–10
7
molecules/cm
3
.
Heterogeneous reactions are extremely efficient, complicating sampling. Measurement
of OH is also challenging because of photolytic interferences. OH is produced in the
atmosphere primarily by the reaction sequence:
O
3
+h
<340 nm
−−−−→
O
2
+O
1
D (4.11)
O
1
D +H
2
O → 2OH (4.12)
An intense light source whose intended purpose is excitation of OH can also photolyze
ozone to produce O
1
D. The false OH concentration is proportional to the square of the
excitation power and linear in ozone and water vapour number density. Early attempts
(Wang & Davis, 1974; Wang et al., 1975, 1976; Davis et al., 1976) to measure tropospheric
OH by LIF detected only laser-generated OH.
The
˜
A
2
+
←X
2
electronic transition of OH is intense and has structured absorption
and fluorescence. The
=0 ←
=0 transition at 308 nm and the
=1 ←
=0
vibronic transition at 282 nm are Doppler-limited and have peak absorption cross sections
of 713 ×10
−15
cm
2
and 143 ×10
−15
cm
2
, respectively (McGee & McIlrath, 1984), and
are used for all measurements of atmospheric OH.
4.3.1 Instruments for stratospheric and upper tropospheric
OH measurements
The first measurements of atmospheric OH using fluorescence were made in 1971 with a
rocket-borne spectrometer in the upper stratosphere and lower mesosphere (Anderson,
1971a,b). These measurements were based on solar-induced fluorescence of OH at twilight
conditions and gave an estimate of the OH column density in the mesosphere.
In 1975 Anderson measured stratospheric OH using an instrument that descended
through the stratosphere while suspended from a stabilized parachute (Anderson, 1976).
Ambient air was sampled as it flowed through the interior of the instrument during
descents from altitudes as high as 43 km. OH was excited in the 0 ← 0 transition at
308 nm and fluorescence was detected at the same wavelength (resonance fluorescence).
The UV light source was a low-pressure microwave discharge of helium mixed with a
small amount of water vapour housed in a quartz tube lamp. A spherical mirror on one
side and a quartz lens on the other were used to produce a collimated beam of photons
corresponding to OH emission lines in the range 306.4–312.0 nm. The lamp flux was
monitored continuously during flight. Fluorescence from excited OH was detected by a
PMT after passing through a narrow-bandpass interference filter centered at 309 nm, a
polarization plate oriented so as to reduce lamp-induced Rayleigh scatter, and a quartz
lens. The instrument background was determined by titration of ambient OH with
HCl injected shortly before the detection point. Resonance fluorescence measurements
of atomic oxygen, O
3
P, were observed using similar instruments (Anderson, 1975;
Takegawa et al., 2001).
200 Analytical Techniques for Atmospheric Measurement
Using this instrument, HO
2
was observed by conversion to OH (Anderson et al., 1981):
HO
2
+NO →OH +NO
2
(4.13)
The conversion efficiency of reaction 4.13 is less than unity due to the three-body reaction
between OH and NO to form nitrous acid (HONO):
OH +NO +M →HONO +M (4.14)
The efficiency of the conversion was quantified by using two or more NO flow rates for
the conversion. The fluorescence observed while the NO is injected stems from both OH
and HO
2
, and so OH measurements (without NO flow) were subtracted to extract HO
2
concentrations.
Despite the success of this instrument it was desirable to have an instrument that would
have higher sensitivity and be capable of more effective tests for interferences. Elimination
of the chemical zero was also desirable. Stimpfle and Anderson (1988) developed a dye
laser pumped by a pulsed copper vapour laser designed to meet these objectives. The
Q
1
1 line at 281.91 nm or the Q
1
2 line at 282.07 nm was pumped, and red-shifted
fluorescence in the 0 ← 0 band at 308 nm was detected. This laser system avoided the
problem of laser-generated OH and maintained high sensitivity by using a high repetition
rate (17 kHz) and a low pulse energy. The laser system was housed in a gondola and
the UV beam was directed into the detection pod where the light was focused into a
multi-pass White cell. The fluorescence was collected by an array of 450 fibre optic cables
and relayed back into the gondola where the fluorescence was delivered to a spectrometer
with a bi-alkali PMT fitted at the exit slit. Nine of the fibre optic cables were directed to
a separate PMT with a 282 nm interference filter in order to measure the Rayleigh scatter
from the optical cell. Observation of this Rayleigh scatter was used to monitor changes
in the sensitivity of the instrument. At a given pressure, any change in the calibration
constant of the instrument (e.g. change in laser alignment or power) will manifest as
an equal change in the observed Rayleigh scatter. Thus, observation of this scatter can
serve as an in-flight proxy calibration and the quantity that must be measured in the
laboratory is the ratio of the instrument’s response to OH and Rayleigh scatter. This
is a design element common to most of the Anderson group’s instruments. The idea
is to find a transfer standard for the instrument calibration that can be accurately and
precisely measured in both the laboratory and under field conditions where thermal
changes, vibrations or other factors result in sub-optimal optical alignment.
The optical collection efficiency of the laser-equipped balloon instrument was subse-
quently improved by installing the PMT directly on the detection pod (Stimpfle et al.,
1990). The spectrometer was replaced by two filters: a chemical solution filter (potassium
hydrogen phthalate dissolved in methyl alcohol) followed by an interference filter (4 nm
full width at half maximum, FWHM, bandpass centred at 310 nm).
Because OH is highly reactive and removed with near-unit efficiency upon collision with
surfaces, sampling requires that the detection volume be unperturbed. This requirement,
which argues for a short, wide, straight sampling tube, is balanced against the requirement
for baffling against solar scatter, which favours the exact opposite – a long, narrow,
bent tube. In the balloon instruments, the sample chamber had a 15 cm diameter, and a
Fluorescence Methods 201
steady descent velocity of ∼5m/s was maintained through radio control of the balloon.
Interaction with the instrument wall was monitored by a thermistor array after the gas
passed through the detection cell. Elevated temperatures near the centre were a diagnostic
for air that had touched the instrument walls and then moved into the detection volume.
Baffles prevented excessive solar scatter in the instrument. The low-altitude limit of
operation was between 23 and 29 km due to either excessive Rayleigh scatter or for, the
laser-equipped instruments, thermal instability of the laser. The 1 uncertainties in the
OH and HO
2
measurements were 18 and 19%, respectively.
The logistical demands of high altitude ballooning limited these instruments to infre-
quent flights of short duration, and consequently in-flight diagnostics were limited.
Also, the feasibility of installing a comprehensive chemical and dynamical payload
on a balloon was low, limiting the potential scientific return. Thus an instrument
that could be mounted on the premier stratospheric chemistry research platform,
NASA’s ER-2, was highly desirable (information on the ER-2 aircraft is available at
http://www.dfrc.nasa.gov/Newsroom/FactSheets/FS-046-DFRC.html). In May 1993, OH
and HO
2
were measured in the lower stratosphere aboard the ER-2 aircraft during the
Stratospheric Photochemistry, Aerosols and Dynamics Expedition (SPADE) (Wennberg
et al., 1994a). A diagram of this instrument is shown in Figure 4.4. This instrument was
much smaller than its balloon-borne predecessor and incorporated an array of diagnostics
that were not possible to implement during the short duration of a balloon flight. The
laser system was a compact <02m
3
<40kg dye laser pumped by two diode-pumped,
6 kHz repetition rate 450 mW Nd:YLF lasers. The conversion efficiency of the dye laser
fundamental into the UV was ∼10%, with a final 282 nm UV beam of linewidth 01cm
−1
.
The instrument was mounted on the nose of the ER-2 aircraft where a sampling system
was devised that allowed controlled flow speeds deep into the instrument without contact
Figure 4.4 The HO
x
instrument mounted in the nose of the NASA ER-2 aircraft. Most of the flow
bypasses the inner duct. A laminar subsection of the flow enters the inner detection duct and passes
through three consecutive OH detection axes. HO
2
is converted to OH by reaction with NO injected
at one of two positions. Water vapour is detected in the third axis (see Section 4.3.3) (From Wennberg
et al., 1995, with permission of American Meteorological Society.)